XOR_compare_directional.ipynb

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      "source": [
        "#@title Default title text\n",
        "\"\"\"Created in May, 2018\n",
        "Pytorch functions for preconditioned SGD\n",
        "@author: XILIN LI, lixilinx@gmail.com\n",
        "@author: OMEAD POOLADZANDI, omeadbpooladzandi@gmail.com\n",
        "\n",
        "Updated in Dec, 2020: \n",
        "Wrapped Kronecker product preconditioner for easy use: the code will select the proper Kronecker product  \n",
        "preconditioner based on the formats of input left and right preconditioners.\n",
        "Add torch.jit.script decorator by default\n",
        "\n",
        "Updated in June, 2022: \n",
        "Added UVd and XMat preconditioner using functional and optim class form.\n",
        "\"\"\"\n",
        "\n",
        "import torch\n",
        "from torch.optim.optimizer import Optimizer\n",
        "\n",
        "###############################################################################\n",
        "@torch.jit.script\n",
        "def update_precond_dense(Q, dxs, dgs, step=0.01, _tiny=1.2e-38):\n",
        "    # type: (Tensor, List[Tensor], List[Tensor], float, float) -> Tensor\n",
        "    \"\"\"\n",
        "    update dense preconditioner P = Q^T*Q\n",
        "    Q: Cholesky factor of preconditioner with positive diagonal entries \n",
        "    dxs: list of perturbations of parameters\n",
        "    dgs: list of perturbations of gradients\n",
        "    step: update step size normalized to range [0, 1] \n",
        "    _tiny: an offset to avoid division by zero \n",
        "    \"\"\"\n",
        "    dx = torch.cat([torch.reshape(x, [-1, 1]) for x in dxs])\n",
        "    dg = torch.cat([torch.reshape(g, [-1, 1]) for g in dgs])\n",
        "    \n",
        "    a = Q.mm(dg)\n",
        "    #b = torch.triangular_solve(dx, Q, upper=True, transpose=True)[0]\n",
        "    b = torch.linalg.solve_triangular(Q.t(), dx, upper=False)\n",
        "\n",
        "    grad = torch.triu(a.mm(a.t()) - b.mm(b.t()))\n",
        "    step0 = step/(grad.abs().max() + _tiny)        \n",
        "        \n",
        "    return Q - step0*grad.mm(Q)\n",
        "\n",
        "@torch.jit.script\n",
        "def precond_grad_dense(Q, grads):\n",
        "    # type: (Tensor, List[Tensor]) -> List[Tensor]\n",
        "    \"\"\"\n",
        "    return preconditioned gradient using dense preconditioner\n",
        "    Q: Cholesky factor of preconditioner\n",
        "    grads: list of gradients\n",
        "    \"\"\"\n",
        "    grad = [torch.reshape(g, [-1, 1]) for g in grads]\n",
        "    lens = [g.shape[0] for g in grad]\n",
        "    grad = torch.cat(grad)\n",
        "    grad = Q.t().mm(Q.mm(grad))\n",
        "    \n",
        "    pre_grads = []\n",
        "    idx = 0\n",
        "    for i in range(len(grads)):\n",
        "        pre_grads.append(torch.reshape(grad[idx : idx + lens[i]], grads[i].shape))\n",
        "        idx = idx + lens[i]\n",
        "        \n",
        "    return pre_grads\n",
        "\n",
        "\n",
        "###############################################################################\n",
        "def update_precond_kron(Ql, Qr, dX, dG, step=0.01, _tiny=1.2e-38):\n",
        "    \"\"\"\n",
        "    Update Kronecker product preconditioner P = kron_prod(Qr^T*Qr, Ql^T*Ql)\n",
        "    Either Ql or Qr can be sparse, and the code can choose the right update rule.\n",
        "    dX: perturbation of (matrix) parameter\n",
        "    dG: perturbation of (matrix) gradient\n",
        "    step: update step size\n",
        "    _tiny: an offset to avoid division by zero \n",
        "    \"\"\"\n",
        "    m, n = Ql.shape\n",
        "    p, q = Qr.shape\n",
        "    if m==n: # left is dense\n",
        "        if p==q: #(dense, dense) format\n",
        "            return _update_precond_dense_dense(Ql, Qr, dX, dG, step, _tiny)\n",
        "        elif p==2: # (dense, normalization) format\n",
        "            return _update_precond_norm_dense(Qr, Ql, dX.t(), dG.t(), step, _tiny)[::-1]\n",
        "        elif p==1: # (dense, scaling) format\n",
        "            return _update_precond_dense_scale(Ql, Qr, dX, dG, step, _tiny)\n",
        "        else:\n",
        "            raise Exception('Unknown Kronecker product preconditioner')\n",
        "    elif m==2: # left is normalization\n",
        "        if p==q: # (normalization, dense) format\n",
        "            return _update_precond_norm_dense(Ql, Qr, dX, dG, step, _tiny)\n",
        "        elif p==1: # (normalization, scaling) format\n",
        "            return _update_precond_norm_scale(Ql, Qr, dX, dG, step, _tiny)\n",
        "        else:\n",
        "            raise Exception('Unknown Kronecker product preconditioner')\n",
        "    elif m==1: # left is scaling\n",
        "        if p==q: # (scaling, dense) format\n",
        "            return _update_precond_dense_scale(Qr, Ql, dX.t(), dG.t(), step, _tiny)[::-1]\n",
        "        elif p==2: # (scaling, normalization) format\n",
        "            return _update_precond_norm_scale(Qr, Ql, dX.t(), dG.t(), step, _tiny)[::-1]\n",
        "        else:\n",
        "            raise Exception('Unknown Kronecker product preconditioner')\n",
        "    else:\n",
        "        raise Exception('Unknown Kronecker product preconditioner')\n",
        " \n",
        "       \n",
        "def precond_grad_kron(Ql, Qr, Grad):\n",
        "    \"\"\"\n",
        "    return preconditioned gradient using Kronecker product preconditioner P = kron_prod(Qr^T*Qr, Ql^T*Ql)\n",
        "    Either Ql or Qr can be sparse, and the code can choose the right way to precondition the gradient\n",
        "    Grad: (matrix) gradient\n",
        "    \"\"\"\n",
        "    m, n = Ql.shape\n",
        "    p, q = Qr.shape\n",
        "    if m==n: # left is dense\n",
        "        if p==q: #(dense, dense) format\n",
        "            return _precond_grad_dense_dense(Ql, Qr, Grad)\n",
        "        elif p==2: # (dense, normalization) format\n",
        "            return _precond_grad_norm_dense(Qr, Ql, Grad.t()).t()\n",
        "        elif p==1: # (dense, scaling) format\n",
        "            return _precond_grad_dense_scale(Ql, Qr, Grad)\n",
        "        else:\n",
        "            raise Exception('Unknown Kronecker product preconditioner')\n",
        "    elif m==2: # left is normalization\n",
        "        if p==q: # (normalization, dense) format\n",
        "            return _precond_grad_norm_dense(Ql, Qr, Grad)\n",
        "        elif p==1: # (normalization, scaling) format\n",
        "            return _precond_grad_norm_scale(Ql, Qr, Grad)\n",
        "        else:\n",
        "            raise Exception('Unknown Kronecker product preconditioner')\n",
        "    elif m==1: # left is scaling\n",
        "        if p==q: # (scaling, dense) format\n",
        "            return _precond_grad_dense_scale(Qr, Ql, Grad.t()).t()\n",
        "        elif p==2: # (scaling, normalization) format\n",
        "            return _precond_grad_norm_scale(Qr, Ql, Grad.t()).t()\n",
        "        else:\n",
        "            raise Exception('Unknown Kronecker product preconditioner')\n",
        "    else:\n",
        "        raise Exception('Unknown Kronecker product preconditioner')\n",
        "        \n",
        "\n",
        "###############################################################################\n",
        "@torch.jit.script\n",
        "def _update_precond_dense_dense(Ql, Qr, dX, dG, step=0.01, _tiny=1.2e-38):\n",
        "    # type: (Tensor, Tensor, Tensor, Tensor, float, float) -> Tuple[Tensor, Tensor]\n",
        "    \"\"\"\n",
        "    update Kronecker product preconditioner P = kron_prod(Qr^T*Qr, Ql^T*Ql)\n",
        "    Ql: (left side) Cholesky factor of preconditioner with positive diagonal entries\n",
        "    Qr: (right side) Cholesky factor of preconditioner with positive diagonal entries\n",
        "    dX: perturbation of (matrix) parameter\n",
        "    dG: perturbation of (matrix) gradient\n",
        "    step: update step size normalized to range [0, 1] \n",
        "    _tiny: an offset to avoid division by zero \n",
        "    \"\"\"\n",
        "    max_l = torch.max(torch.diag(Ql))\n",
        "    max_r = torch.max(torch.diag(Qr))\n",
        "    \n",
        "    rho = torch.sqrt(max_l/max_r)\n",
        "    Ql /= rho\n",
        "    Qr *= rho\n",
        "    \n",
        "    #A = Ql.mm( dG.mm( Qr.t() ) )\n",
        "    #Bt = torch.triangular_solve((torch.triangular_solve(dX.t(), Qr, upper=True, transpose=True))[0].t(), \n",
        "    #                 Ql, upper=True, transpose=True)[0]\n",
        "    A = torch.linalg.multi_dot([Ql, dG, Qr.t()])\n",
        "    Bt = torch.linalg.solve_triangular(Ql.t(), torch.linalg.solve_triangular(Qr, dX, upper=True, left=False), upper=False)\n",
        "    \n",
        "    grad1 = torch.triu(A.mm(A.t()) - Bt.mm(Bt.t()))\n",
        "    grad2 = torch.triu(A.t().mm(A) - Bt.t().mm(Bt))\n",
        "    \n",
        "    step1 = step/(torch.max(torch.abs(grad1)) + _tiny)\n",
        "    step2 = step/(torch.max(torch.abs(grad2)) + _tiny)\n",
        "        \n",
        "    return Ql - step1*grad1.mm(Ql), Qr - step2*grad2.mm(Qr)\n",
        "    \n",
        "@torch.jit.script\n",
        "def _precond_grad_dense_dense(Ql, Qr, Grad):\n",
        "    # type: (Tensor, Tensor, Tensor) -> Tensor\n",
        "    \"\"\"\n",
        "    return preconditioned gradient using Kronecker product preconditioner\n",
        "    Ql: (left side) Cholesky factor of preconditioner\n",
        "    Qr: (right side) Cholesky factor of preconditioner\n",
        "    Grad: (matrix) gradient\n",
        "    \"\"\"\n",
        "    #return torch.chain_matmul(Ql.t(), Ql, Grad, Qr.t(), Qr)\n",
        "    return torch.linalg.multi_dot([Ql.t(), Ql, Grad, Qr.t(), Qr])\n",
        "    \n",
        "\n",
        "###############################################################################\n",
        "# (normalization, dense) format Kronecker product preconditioner\n",
        "@torch.jit.script\n",
        "def _update_precond_norm_dense(ql, Qr, dX, dG, step=0.01, _tiny=1.2e-38):\n",
        "    # type: (Tensor, Tensor, Tensor, Tensor, float, float) -> Tuple[Tensor, Tensor]\n",
        "    \"\"\"\n",
        "    update (normalization, dense) Kronecker product preconditioner P = kron_prod(Qr^T*Qr, Ql^T*Ql), where\n",
        "    dX and dG have shape (M, N)\n",
        "    ql has shape (2, M)\n",
        "    Qr has shape (N, N)\n",
        "    ql[0] is the diagonal part of Ql\n",
        "    ql[1,0:-1] is the last column of Ql, excluding the last entry\n",
        "    dX is perturbation of (matrix) parameter\n",
        "    dG is perturbation of (matrix) gradient\n",
        "    step: update step size normalized to range [0, 1] \n",
        "    _tiny: an offset to avoid division by zero  \n",
        "    \"\"\"\n",
        "    # make sure that Ql and Qr have similar dynamic range\n",
        "    max_l = torch.max(ql[0])\n",
        "    max_r = torch.max(torch.diag(Qr))  \n",
        "    rho = torch.sqrt(max_l/max_r)\n",
        "    ql /= rho\n",
        "    Qr *= rho\n",
        "    \n",
        "    # refer to https://arxiv.org/abs/1512.04202 for details\n",
        "    A = ql[0:1].t()*dG + ql[1:].t().mm( dG[-1:] ) # Ql*dG \n",
        "    A = A.mm(Qr.t())\n",
        "    \n",
        "    Bt = dX/ql[0:1].t()\n",
        "    Bt[-1:] -= (ql[1:]/(ql[0:1]*ql[0,-1])).mm(dX)\n",
        "    #Bt = torch.triangular_solve(Bt.t(), Qr, upper=True, transpose=True)[0].t()\n",
        "    Bt = torch.linalg.solve_triangular(Qr, Bt, upper=True, left=False)\n",
        "    \n",
        "    grad1_diag = torch.sum(A*A, dim=1) - torch.sum(Bt*Bt, dim=1)\n",
        "    grad1_bias = A[:-1].mm(A[-1:].t()) - Bt[:-1].mm(Bt[-1:].t()) \n",
        "    grad1_bias = torch.cat([torch.squeeze(grad1_bias), grad1_bias.new_zeros(1)])  \n",
        "\n",
        "    step1 = step/(torch.max(torch.max(torch.abs(grad1_diag)), \n",
        "                            torch.max(torch.abs(grad1_bias))) + _tiny)\n",
        "    new_ql0 = ql[0] - step1*grad1_diag*ql[0]\n",
        "    new_ql1 = ql[1] - step1*(grad1_diag*ql[1] + ql[0,-1]*grad1_bias)\n",
        "    \n",
        "    grad2 = torch.triu(A.t().mm(A) - Bt.t().mm(Bt))\n",
        "    step2 = step/(torch.max(torch.abs(grad2)) + _tiny)\n",
        "    \n",
        "    return torch.stack((new_ql0, new_ql1)), Qr - step2*grad2.mm(Qr)\n",
        "\n",
        "@torch.jit.script\n",
        "def _precond_grad_norm_dense(ql, Qr, Grad):\n",
        "    # type: (Tensor, Tensor, Tensor) -> Tensor\n",
        "    \"\"\"\n",
        "    return preconditioned gradient using (normalization, dense) Kronecker product preconditioner \n",
        "    Suppose Grad has shape (M, N)\n",
        "    ql[0] is the diagonal part of Ql\n",
        "    ql[1, 0:-1] is the last column of Ql, excluding the last entry\n",
        "    Qr: shape (N, N), Cholesky factor of right preconditioner\n",
        "    Grad: (matrix) gradient\n",
        "    \"\"\"\n",
        "    preG = ql[0:1].t()*Grad + ql[1:].t().mm(Grad[-1:]) # Ql*Grad \n",
        "    #preG = torch.chain_matmul(preG, Qr.t(), Qr)\n",
        "    preG = torch.linalg.multi_dot([preG, Qr.t(), Qr])\n",
        "    add_last_row = ql[1:].mm(preG) # use it to modify the last row\n",
        "    preG *= ql[0:1].t()\n",
        "    preG[-1:] += add_last_row\n",
        "    \n",
        "    return preG\n",
        "\n",
        "\n",
        "###############################################################################\n",
        "# (normalization, scaling) Kronecker product preconditioner \n",
        "# the left one is a normalization preconditioner; the right one is a scaling preconditioner\n",
        "@torch.jit.script\n",
        "def _update_precond_norm_scale(ql, qr, dX, dG, step=0.01, _tiny=1.2e-38):\n",
        "    # type: (Tensor, Tensor, Tensor, Tensor, float, float) -> Tuple[Tensor, Tensor]\n",
        "    \"\"\"\n",
        "    update (normalization, scaling) preconditioner P = kron_prod(Qr^T*Qr, Ql^T*Ql), where\n",
        "    dX and dG have shape (M, N)\n",
        "    ql has shape (2, M)\n",
        "    qr has shape (1, N)\n",
        "    ql[0] is the diagonal part of Ql\n",
        "    ql[1, 0:-1] is the last column of Ql, excluding the last entry\n",
        "    qr is the diagonal part of Qr\n",
        "    dX is perturbation of (matrix) parameter\n",
        "    dG is perturbation of (matrix) gradient\n",
        "    step: update step size\n",
        "    _tiny: an offset to avoid division by zero  \n",
        "    \"\"\"\n",
        "    # make sure that Ql and Qr have similar dynamic range\n",
        "    max_l = torch.max(ql[0])\n",
        "    max_r = torch.max(qr) # qr always is positive\n",
        "    rho = torch.sqrt(max_l/max_r)\n",
        "    ql /= rho\n",
        "    qr *= rho\n",
        "    \n",
        "    # refer to https://arxiv.org/abs/1512.04202 for details\n",
        "    A = ql[0:1].t()*dG + ql[1:].t().mm( dG[-1:] ) # Ql*dG \n",
        "    A *= qr # Ql*dG*Qr \n",
        "    \n",
        "    Bt = dX/ql[0:1].t()\n",
        "    Bt[-1:] -= (ql[1:]/(ql[0:1]*ql[0,-1])).mm(dX)\n",
        "    Bt /= qr # Ql^(-T)*dX*Qr^(-1) \n",
        "    \n",
        "    grad1_diag = torch.sum(A*A, dim=1) - torch.sum(Bt*Bt, dim=1)\n",
        "    grad1_bias = A[:-1].mm(A[-1:].t()) - Bt[:-1].mm(Bt[-1:].t()) \n",
        "    grad1_bias = torch.cat([torch.squeeze(grad1_bias), grad1_bias.new_zeros(1)])  \n",
        "\n",
        "    step1 = step/(torch.max(torch.max(torch.abs(grad1_diag)), \n",
        "                            torch.max(torch.abs(grad1_bias))) + _tiny)\n",
        "    new_ql0 = ql[0] - step1*grad1_diag*ql[0]\n",
        "    new_ql1 = ql[1] - step1*(grad1_diag*ql[1] + ql[0,-1]*grad1_bias)\n",
        "    \n",
        "    grad2 = torch.sum(A*A, dim=0, keepdim=True) - torch.sum(Bt*Bt, dim=0, keepdim=True)\n",
        "    step2 = step/(torch.max(torch.abs(grad2)) + _tiny)\n",
        "    \n",
        "    return torch.stack((new_ql0, new_ql1)), qr - step2*grad2*qr\n",
        "\n",
        "@torch.jit.script\n",
        "def _precond_grad_norm_scale(ql, qr, Grad):\n",
        "    # type: (Tensor, Tensor, Tensor) -> Tensor\n",
        "    \"\"\"\n",
        "    return preconditioned gradient using (normalization, scaling) Kronecker product preconditioner\n",
        "    Suppose Grad has shape (M, N)\n",
        "    ql has shape (2, M) \n",
        "    qr has shape (1, N) \n",
        "    ql[0] is the diagonal part of Ql\n",
        "    ql[1, 0:-1] is the last column of Ql, excluding the last entry\n",
        "    qr is the diagonal part of Qr\n",
        "    Grad: (matrix) gradient\n",
        "    \"\"\"\n",
        "    preG = ql[0:1].t()*Grad + ql[1:].t().mm(Grad[-1:]) # Ql*Grad \n",
        "    preG *= (qr*qr) # Ql*Grad*Qr^T*Qr\n",
        "    add_last_row = ql[1:].mm(preG) # use it to modify the last row\n",
        "    preG *= ql[0:1].t()\n",
        "    preG[-1:] += add_last_row\n",
        "    \n",
        "    return preG\n",
        "\n",
        "\n",
        "###############################################################################\n",
        "@torch.jit.script\n",
        "def _update_precond_dense_scale(Ql, qr, dX, dG, step=0.01, _tiny=1.2e-38):\n",
        "    # type: (Tensor, Tensor, Tensor, Tensor, float, float) -> Tuple[Tensor, Tensor]\n",
        "    \"\"\"\n",
        "    update (dense, scaling) preconditioner P = kron_prod(Qr^T*Qr, Ql^T*Ql), where\n",
        "    dX and dG have shape (M, N)\n",
        "    Ql has shape (M, M)\n",
        "    qr has shape (1, N)\n",
        "    qr is the diagonal part of Qr\n",
        "    dX is perturbation of (matrix) parameter\n",
        "    dG is perturbation of (matrix) gradient\n",
        "    step: update step size\n",
        "    _tiny: an offset to avoid division by zero \n",
        "    \"\"\"\n",
        "    max_l = torch.max(torch.diag(Ql))\n",
        "    max_r = torch.max(qr)\n",
        "    \n",
        "    rho = torch.sqrt(max_l/max_r)\n",
        "    Ql /= rho\n",
        "    qr *= rho\n",
        "    \n",
        "    A = Ql.mm( dG*qr )\n",
        "    #Bt = torch.triangular_solve(dX/qr, Ql, upper=True, transpose=True)[0]\n",
        "    Bt = torch.linalg.solve_triangular(Ql.t(), dX/qr, upper=False)\n",
        "    \n",
        "    grad1 = torch.triu(A.mm(A.t()) - Bt.mm(Bt.t()))\n",
        "    grad2 = torch.sum(A*A, dim=0, keepdim=True) - torch.sum(Bt*Bt, dim=0, keepdim=True)\n",
        "    \n",
        "    step1 = step/(torch.max(torch.abs(grad1)) + _tiny)\n",
        "    step2 = step/(torch.max(torch.abs(grad2)) + _tiny)\n",
        "        \n",
        "    return Ql - step1*grad1.mm(Ql), qr - step2*grad2*qr\n",
        "    \n",
        "@torch.jit.script\n",
        "def _precond_grad_dense_scale(Ql, qr, Grad):\n",
        "    # type: (Tensor, Tensor, Tensor) -> Tensor\n",
        "    \"\"\"\n",
        "    return preconditioned gradient using (dense, scaling) Kronecker product preconditioner\n",
        "    Suppose Grad has shape (M, N)\n",
        "    Ql: shape (M, M), (left side) Cholesky factor of preconditioner\n",
        "    qr: shape (1, N), defines a diagonal matrix for output feature scaling\n",
        "    Grad: (matrix) gradient\n",
        "    \"\"\"\n",
        "    #return torch.chain_matmul(Ql.t(), Ql, Grad*(qr*qr))\n",
        "    return torch.linalg.multi_dot([Ql.t(), Ql, Grad*(qr*qr)])\n",
        "\n",
        "\n",
        "\n",
        "###############################################################################   \n",
        "@torch.jit.script                     \n",
        "def update_precond_splu(L12, l3, U12, u3, dxs, dgs, step=0.01, _tiny=1.2e-38):\n",
        "    # type: (Tensor,Tensor,Tensor,Tensor, List[Tensor],List[Tensor], float,float) -> Tuple[Tensor,Tensor,Tensor,Tensor]\n",
        "    \"\"\"\n",
        "    update sparse LU preconditioner P = Q^T*Q, where \n",
        "    Q = L*U,\n",
        "    L12 = [L1; L2]\n",
        "    U12 = [U1, U2]\n",
        "    L = [L1, 0; L2, diag(l3)]\n",
        "    U = [U1, U2; 0, diag(u3)]\n",
        "    l3 and u3 are column vectors\n",
        "    dxs: a list of random perturbation on parameters\n",
        "    dgs: a list of resultant perturbation on gradients\n",
        "    step: update step size normalized to range [0, 1] \n",
        "    _tiny: an offset to avoid division by zero \n",
        "    \"\"\"\n",
        "    # make sure that L and U have similar dynamic range\n",
        "    max_l = torch.max(torch.max(torch.diag(L12)), torch.max(l3))\n",
        "    max_u = torch.max(torch.max(torch.diag(U12)), torch.max(u3))\n",
        "    rho = torch.sqrt(max_l/max_u)\n",
        "    L12 /= rho\n",
        "    l3 /= rho\n",
        "    U12 *= rho\n",
        "    u3 *= rho\n",
        "    \n",
        "    # extract the blocks\n",
        "    r = U12.shape[0]\n",
        "    L1 = L12[:r]\n",
        "    L2 = L12[r:]\n",
        "    U1 = U12[:, :r]\n",
        "    U2 = U12[:, r:]\n",
        "    \n",
        "    dx = torch.cat([torch.reshape(x, [-1, 1]) for x in dxs]) # a tall column vector\n",
        "    dg = torch.cat([torch.reshape(g, [-1, 1]) for g in dgs]) # a tall column vector\n",
        "    \n",
        "    # U*dg\n",
        "    Ug1 = U1.mm(dg[:r]) + U2.mm(dg[r:])\n",
        "    Ug2 = u3*dg[r:]\n",
        "    # Q*dg\n",
        "    Qg1 = L1.mm(Ug1)\n",
        "    Qg2 = L2.mm(Ug1) + l3*Ug2\n",
        "    # inv(U^T)*dx\n",
        "    #iUtx1 = torch.triangular_solve(dx[:r], U1, upper=True, transpose=True)[0]\n",
        "    iUtx1 = torch.linalg.solve_triangular(U1.t(), dx[:r], upper=False)\n",
        "    iUtx2 = (dx[r:] - U2.t().mm(iUtx1))/u3\n",
        "    # inv(Q^T)*dx\n",
        "    iQtx2 = iUtx2/l3\n",
        "    #iQtx1 = torch.triangular_solve(iUtx1 - L2.t().mm(iQtx2), L1, upper=False, transpose=True)[0]\n",
        "    iQtx1 = torch.linalg.solve_triangular(L1.t(), iUtx1 - L2.t().mm(iQtx2), upper=True)\n",
        "    # L^T*Q*dg\n",
        "    LtQg1 = L1.t().mm(Qg1) + L2.t().mm(Qg2)\n",
        "    LtQg2 = l3*Qg2\n",
        "    # P*dg\n",
        "    Pg1 = U1.t().mm(LtQg1)\n",
        "    Pg2 = U2.t().mm(LtQg1) + u3*LtQg2\n",
        "    # inv(L)*inv(Q^T)*dx\n",
        "    #iLiQtx1 = torch.triangular_solve(iQtx1, L1, upper=False)[0]\n",
        "    iLiQtx1 = torch.linalg.solve_triangular(L1, iQtx1, upper=False)\n",
        "    iLiQtx2 = (iQtx2 - L2.mm(iLiQtx1))/l3\n",
        "    # inv(P)*dx\n",
        "    iPx2 = iLiQtx2/u3\n",
        "    #iPx1 = torch.triangular_solve(iLiQtx1 - U2.mm(iPx2), U1, upper=True)[0]\n",
        "    iPx1 = torch.linalg.solve_triangular(U1, iLiQtx1 - U2.mm(iPx2), upper=True)\n",
        "    \n",
        "    # update L\n",
        "    grad1 = Qg1.mm(Qg1.t()) - iQtx1.mm(iQtx1.t())\n",
        "    grad1 = torch.tril(grad1)\n",
        "    grad2 = Qg2.mm(Qg1.t()) - iQtx2.mm(iQtx1.t())\n",
        "    grad3 = Qg2*Qg2 - iQtx2*iQtx2\n",
        "    max_abs_grad = torch.max(torch.abs(grad1))\n",
        "    max_abs_grad = torch.max(max_abs_grad, torch.max(torch.abs(grad2)))\n",
        "    max_abs_grad = torch.max(max_abs_grad, torch.max(torch.abs(grad3)))\n",
        "    step0 = step/(max_abs_grad + _tiny)\n",
        "    newL1 = L1 - step0*grad1.mm(L1)\n",
        "    newL2 = L2 - step0*grad2.mm(L1) - step0*grad3*L2\n",
        "    newl3 = l3 - step0*grad3*l3\n",
        "\n",
        "    # update U\n",
        "    grad1 = Pg1.mm(dg[:r].t()) - dx[:r].mm(iPx1.t())\n",
        "    grad1 = torch.triu(grad1)\n",
        "    grad2 = Pg1.mm(dg[r:].t()) - dx[:r].mm(iPx2.t())\n",
        "    grad3 = Pg2*dg[r:] - dx[r:]*iPx2\n",
        "    max_abs_grad = torch.max(torch.abs(grad1))\n",
        "    max_abs_grad = torch.max(max_abs_grad, torch.max(torch.abs(grad2)))\n",
        "    max_abs_grad = torch.max(max_abs_grad, torch.max(torch.abs(grad3)))\n",
        "    step0 = step/(max_abs_grad + _tiny)\n",
        "    newU1 = U1 - U1.mm(step0*grad1)\n",
        "    newU2 = U2 - U1.mm(step0*grad2) - step0*grad3.t()*U2\n",
        "    newu3 = u3 - step0*grad3*u3\n",
        "\n",
        "    return torch.cat([newL1, newL2], dim=0), newl3, torch.cat([newU1, newU2], dim=1), newu3\n",
        "\n",
        "@torch.jit.script\n",
        "def precond_grad_splu(L12, l3, U12, u3, grads):\n",
        "    # type: (Tensor,Tensor,Tensor,Tensor, List[Tensor]) -> List[Tensor]\n",
        "    \"\"\"\n",
        "    return preconditioned gradient with sparse LU preconditioner\n",
        "    where P = Q^T*Q, \n",
        "    Q = L*U,\n",
        "    L12 = [L1; L2]\n",
        "    U12 = [U1, U2]\n",
        "    L = [L1, 0; L2, diag(l3)]\n",
        "    U = [U1, U2; 0, diag(u3)]\n",
        "    l3 and u3 are column vectors\n",
        "    grads: a list of gradients to be preconditioned\n",
        "    \"\"\"\n",
        "    grad = [torch.reshape(g, [-1, 1]) for g in grads] # a list of column vector\n",
        "    lens = [g.shape[0] for g in grad] # length of each column vector\n",
        "    grad = torch.cat(grad)  # a tall column vector\n",
        "    \n",
        "    r = U12.shape[0]\n",
        "    L1 = L12[:r]\n",
        "    L2 = L12[r:]\n",
        "    U1 = U12[:, :r]\n",
        "    U2 = U12[:, r:]    \n",
        "    \n",
        "    # U*g\n",
        "    Ug1 = U1.mm(grad[:r]) + U2.mm(grad[r:])\n",
        "    Ug2 = u3*grad[r:]\n",
        "    # Q*g\n",
        "    Qg1 = L1.mm(Ug1)\n",
        "    Qg2 = L2.mm(Ug1) + l3*Ug2\n",
        "    # L^T*Q*g\n",
        "    LtQg1 = L1.t().mm(Qg1) + L2.t().mm(Qg2)\n",
        "    LtQg2 = l3*Qg2\n",
        "    # P*g\n",
        "    pre_grad = torch.cat([U1.t().mm(LtQg1),\n",
        "                          U2.t().mm(LtQg1) + u3*LtQg2])\n",
        "    \n",
        "    pre_grads = [] # restore pre_grad to its original shapes\n",
        "    idx = 0\n",
        "    for i in range(len(grads)):\n",
        "        pre_grads.append(torch.reshape(pre_grad[idx : idx + lens[i]], grads[i].shape))\n",
        "        idx = idx + lens[i]\n",
        "    \n",
        "    return pre_grads\n",
        "\n",
        "\n",
        "\n",
        "##############################################################################\n",
        "#\n",
        "# The low-rank approximation (UVd) preconditioner is defined by\n",
        "#\n",
        "#   Q = (I + U*V')*diag(d)\n",
        "#\n",
        "# which, after reparameterization, is equivalent to form\n",
        "#\n",
        "#   diag(d) + U*V'\n",
        "# \n",
        "# It relates to the LM-BFGS and conjugate gradient methods. \n",
        "#\n",
        "# The JIT decorator can be enabled if helps. \n",
        "# \n",
        "\n",
        "#@torch.jit.script\n",
        "def IpUVtmatvec(U, V, x):\n",
        "    # type: (Tensor, Tensor, Tensor) -> Tensor\n",
        "    \"\"\"\n",
        "    Returns (I + U*V')*x. All variables are either matrices or column vectors. \n",
        "    \"\"\"\n",
        "    return x + U.mm(V.t().mm(x))\n",
        "\n",
        "# def IpUVtsolve(U, V, x):\n",
        "#     \"\"\"\n",
        "#     Returns inv(I + U*V')*x. All variables are either matrices or column vectors.\n",
        "#     \"\"\"\n",
        "#     VtU = V.t().mm(U)\n",
        "#     I = torch.eye(VtU.size(dim=0), dtype=VtU.dtype, device=VtU.device)\n",
        "#     return x - U.mm(torch.linalg.solve(I + VtU, V.t().mm(x))) # torch.solve is slow\n",
        "\n",
        "# def norm_UVt(U, V):\n",
        "#     \"\"\"\n",
        "#     Returns ||U*V'||_fro = sqrt(tr(U'*U*V'*V)) = sqrt(sum((U'*U)*(V'*V))) \n",
        "#     \"\"\"\n",
        "#     return torch.sqrt(torch.abs(torch.sum( (U.t().mm(U))*(V.t().mm(V)) )))\n",
        "\n",
        "#@torch.jit.script\n",
        "def update_precond_UVd_math_(U, V, d, v, h, step, tiny):\n",
        "    # type: (Tensor, Tensor, Tensor, Tensor, Tensor, float, float) -> None\n",
        "    \"\"\"\n",
        "    Update preconditioner Q = (I + U*V')*diag(d) with (vector, Hessian-vector product) = (v, h).\n",
        "    State variables U, V and d are updated inplace. \n",
        "                               \n",
        "    U, V, d, v, and h are either matrices or column vectors.  \n",
        "    \"\"\"\n",
        "    # balance the numerical dynamic ranges of U and V; optional \n",
        "    if torch.rand([]) < 0.01:\n",
        "        normU = torch.linalg.vector_norm(U)\n",
        "        normV = torch.linalg.vector_norm(V)\n",
        "        rho = torch.sqrt(normU/normV)\n",
        "        U.div_(rho)\n",
        "        V.mul_(rho)\n",
        "\n",
        "    Qh = IpUVtmatvec(U, V, d*h)\n",
        "    Ph = d*IpUVtmatvec(V, U, Qh)\n",
        "    \n",
        "    # invQtv = IpUVtsolve(V, U, v/d)\n",
        "    # invPv = IpUVtsolve(U, V, invQtv)/d\n",
        "    VtU = V.t().mm(U)\n",
        "    I = torch.eye(VtU.size(dim=0), dtype=VtU.dtype, device=VtU.device)\n",
        "    IpVtU = I + VtU\n",
        "    invQtv = v/d\n",
        "    # torch's linalg.solve is slow for small matrix\n",
        "    invQtv = invQtv - V.mm(torch.linalg.solve(IpVtU.t(), U.t().mm(invQtv)))  \n",
        "    invPv  = invQtv - U.mm(torch.linalg.solve(IpVtU,     V.t().mm(invQtv)))\n",
        "    invPv = invPv/d\n",
        "\n",
        "    nablaD = Ph*h - v*invPv\n",
        "    mu = step/(torch.max(torch.abs(nablaD)) + tiny)\n",
        "    #d = d - mu*d*nablaD\n",
        "    d.sub_(mu*d*nablaD)\n",
        "    \n",
        "    # update either U or V, not both at the same time\n",
        "    a, b = Qh, invQtv\n",
        "    if torch.rand([]) < 0.5:\n",
        "        # nablaU = Qh.mm(Qh.t().mm(V)) - invQtv.mm(invQtv.t().mm(V))\n",
        "        # mu = step/(norm_UVt(nablaU, V) + _tiny)\n",
        "        # U = U - mu*(nablaU + nablaU.mm(V.t().mm(U)))\n",
        "        atV = a.t().mm(V)\n",
        "        atVVt = atV.mm(V.t())\n",
        "        btV = b.t().mm(V)\n",
        "        btVVt = btV.mm(V.t())\n",
        "        norm = torch.sqrt(torch.abs( (a.t().mm(a))*(atVVt.mm(atVVt.t())) # abs to avoid sqrt(-0.0) \n",
        "                                    +(b.t().mm(b))*(btVVt.mm(btVVt.t())) \n",
        "                                  -2*(a.t().mm(b))*(atVVt.mm(btVVt.t())) ))\n",
        "        mu = step/(norm + tiny)\n",
        "        # U = U - mu*( a.mm(atV.mm(IpVtU)) \n",
        "        #             -b.mm(btV.mm(IpVtU)) )\n",
        "        U.sub_(mu*( a.mm(atV.mm(IpVtU)) \n",
        "                   -b.mm(btV.mm(IpVtU)) ))\n",
        "    else:\n",
        "        # nablaV = Qh.mm(Qh.t().mm(U)) - invQtv.mm(invQtv.t().mm(U))\n",
        "        # mu = step/(norm_UVt(U, nablaV) + _tiny)\n",
        "        # V = V - mu*(nablaV + V.mm(U.t().mm(nablaV)))\n",
        "        atU = a.t().mm(U)\n",
        "        btU = b.t().mm(U)\n",
        "        UUta = U.mm(atU.t())\n",
        "        UUtb = U.mm(btU.t())\n",
        "        norm = torch.sqrt(torch.abs( (UUta.t().mm(UUta))*(a.t().mm(a)) # abs to avoid sqrt(-0.0)\n",
        "                                    +(UUtb.t().mm(UUtb))*(b.t().mm(b))\n",
        "                                  -2*(UUta.t().mm(UUtb))*(a.t().mm(b)) ))\n",
        "        mu = step/(norm + tiny)\n",
        "        # V = V - mu*( (a + V.mm(atU.t())).mm(atU) \n",
        "        #             -(b + V.mm(btU.t())).mm(btU) )\n",
        "        V.sub_(mu*( (a + V.mm(atU.t())).mm(atU) \n",
        "                   -(b + V.mm(btU.t())).mm(btU) ))\n",
        "\n",
        "    # return [U, V, d]\n",
        "\n",
        "#@torch.jit.script\n",
        "def precond_grad_UVd_math(U, V, d, g):\n",
        "    # type: (Tensor, Tensor, Tensor, Tensor) -> Tensor\n",
        "    \"\"\"\n",
        "    Preconditioning gradient g with Q = (I + U*V')*diag(d).\n",
        "                                         \n",
        "    All variables here are either matrices or column vectors. \n",
        "    \"\"\"\n",
        "    g = IpUVtmatvec(U, V, d*g)\n",
        "    g = d*IpUVtmatvec(V, U, g)\n",
        "    return g\n",
        "\n",
        "## Functional form of UVd Precond\n",
        "\n",
        "def update_precond_UVd(UVd, vs, hs, step=0.01, _tiny=1.2e-38):\n",
        "    # type: (Tensor, List[Tensor], List[Tensor], float, float) -> Tensor\n",
        "    \"\"\"\n",
        "    update UVd preconditioner Q = (I + U*V')*diag(d) with\n",
        "    vs: a list of vectors;\n",
        "    hs: a list of associated Hessian-vector products;\n",
        "    step: step size, setting to larger values, say 0.1, if updating is sparse;\n",
        "    _tiny: an offset to avoid divided by zero. \n",
        "    \n",
        "    It is a wrapped version of function update_precond_UVd_math for easy use. \n",
        "    Also, U, V, and d are transposed (row-major order as Python convention), and \n",
        "    packaged into one tensor. \n",
        "    \"\"\" \n",
        "    sizes = [len(UVd)//2]*2 + [1]\n",
        "    U, V, d = torch.split(UVd.t(), sizes, dim=1)\n",
        "\n",
        "    v = torch.cat([torch.flatten(v) for v in vs])\n",
        "    h = torch.cat([torch.flatten(h) for h in hs])\n",
        "    update_precond_UVd_math_(U, V, d, v[:,None], h[:,None], step=step, tiny=_tiny)\n",
        "    return torch.cat([U, V, d], 1).t()\n",
        "\n",
        "\n",
        "#@torch.jit.script\n",
        "def precond_grad_UVd(UVd, grads):\n",
        "    # type: (Tensor, List[Tensor]) -> List[Tensor]\n",
        "    \"\"\"\n",
        "    return preconditioned gradient with UVd preconditioner Q = (I + U*V')*diag(d),\n",
        "    and a list of gradients, grads.\n",
        "    \n",
        "    It is a wrapped version of function precond_grad_UVd_math for easy use.\n",
        "    Also, U, V, and d are transposed (row-major order as Python convention), and \n",
        "    packaged into one tensor.\n",
        "    \"\"\"\n",
        "    sizes = [len(UVd)//2]*2 + [1]\n",
        "    U, V, d = torch.split(UVd.t(), sizes, dim=1)\n",
        "\n",
        "    # record the sizes and shapes, and then flatten gradients\n",
        "    sizes = [torch.numel(g) for g in grads]\n",
        "    shapes = [g.shape for g in grads]\n",
        "    cumsizes = torch.cumsum(torch.tensor(sizes), 0)\n",
        "    \n",
        "    grad = torch.cat([torch.flatten(g) for g in grads])\n",
        "\n",
        "    # precondition gradients\n",
        "    pre_grad = precond_grad_UVd_math(U, V, d, grad[:,None])\n",
        "\n",
        "    # restore gradients to their original shapes\n",
        "    return [torch.reshape(pre_grad[j-i:j], s) for (i, j, s) in zip(sizes, cumsizes, shapes)]\n",
        "\n",
        "class UVd(Optimizer):\n",
        "    \"\"\"\n",
        "    Implements the low-rank approximation (UVd) preconditioner, Q = (I + U*V')*diag(d), as a class.\n",
        "\n",
        "    Args for initialization:\n",
        "        params_with_grad: a list of parameters or variables requiring gradients;\n",
        "        rank_of_approximation: rank of approximation, i.e., rank of U or V;\n",
        "        preconditioner_init_scale: initial scale of Q, or roughly, Q = preconditioner_init_scale*eye();\n",
        "        lr_params: normalized learning rate for parameters in range [0, 1];\n",
        "        lr_preconditioner: normalized learning rate for preconditioner in range [0, 1];\n",
        "        momentum: momentum factor in range [0,1);\n",
        "        grad_clip_max_norm: maximum allowable gradient norm after clipping, None for no clipping;\n",
        "        preconditioner_update_probability: probability on updating Q, 1 for updating at every step, and 0 for never;\n",
        "        exact_hessian_vector_product: True for exact Hessian-vector product via 2nd derivative,\n",
        "                                    and False for approximate one via finite-difference formulae.\n",
        "\n",
        "    Notes:\n",
        "        Note 1: The Hessian-vector product can be approximated using the finite-difference formulae by setting \n",
        "        exact_hessian_vector_product = False when the 2nd derivatives is not available.\n",
        "        In this case, make sure that the closure produces the same outputs given the same inputs, \n",
        "        except for numerical errors due to non-deterministic behaviors.\n",
        "        Random numbers, if any, used inside the closure should be generated starting from the same state, where the rng state can be\n",
        "        read and set by, e.g., `torch.cuda.get_rng_state' and `torch.cuda.set_rng_state', respectively.\n",
        "        \n",
        "        Note 2: Momentum here is the moving average of gradient so that its setting is decoupled from the learning rate.\n",
        "        This is necessary as the learning rate in PSGD is normalized. \n",
        "\n",
        "        Note 3: `torch.linalg.solve' is called twice in function `update_precond_UVd_math_'.\n",
        "        Certain solver could be orders of magnitude faster than others, especially for small matrices (see the pdf file).\n",
        "        Considering replace it with faster ones if the default solver is too slow.\n",
        "\n",
        "        Note 4: Currently, no support of sparse and mixed-precision gradients. \n",
        "        Half precision is supported except that torch.linalg.solve (v1.12) requires casting float16 to float32.    \n",
        "        \n",
        "        Note 5: lr_params, lr_preconditioner, momentum, grad_clip_max_norm, preconditioner_update_probability, and \n",
        "        exact_hessian_vector_product (bool) all can be reset on the fly. \n",
        "    \"\"\"\n",
        "    def __init__(self,  params_with_grad, rank_of_approximation:int=10, preconditioner_init_scale=1.0,\n",
        "                        lr_params=0.01, lr_preconditioner=0.01, momentum=0.0,\n",
        "                        grad_clip_max_norm=None, preconditioner_update_probability=1.0,\n",
        "                        exact_hessian_vector_product:bool=True, directional = 1):\n",
        "        # mutable members\n",
        "        self.lr_params = lr_params\n",
        "        self.lr_preconditioner = lr_preconditioner\n",
        "        self.momentum = momentum if (0<momentum<1) else 0.0\n",
        "        self.grad_clip_max_norm = grad_clip_max_norm\n",
        "        self.preconditioner_update_probability = preconditioner_update_probability\n",
        "        self.exact_hessian_vector_product = exact_hessian_vector_product\n",
        "        self.directional = directional\n",
        "        # protected members\n",
        "        params_with_grad = [params_with_grad,] if isinstance(params_with_grad, torch.Tensor) else params_with_grad\n",
        "        self._params_with_grad = [param for param in params_with_grad if param.requires_grad] # double check requires_grad flag\n",
        "        dtype, device = self._params_with_grad[0].dtype, self._params_with_grad[0].device\n",
        "        self._tiny = torch.finfo(dtype).tiny\n",
        "        self._delta_param_scale = torch.finfo(dtype).eps**0.5\n",
        "        self._param_sizes = [torch.numel(param) for param in self._params_with_grad]\n",
        "        self._param_cumsizes = torch.cumsum(torch.tensor(self._param_sizes), 0)\n",
        "        num_params = self._param_cumsizes[-1]\n",
        "        self._U = torch.randn(num_params, rank_of_approximation, dtype=dtype, device=device) / (num_params*rank_of_approximation)**0.5\n",
        "        self._V = torch.randn(num_params, rank_of_approximation, dtype=dtype, device=device) / (num_params*rank_of_approximation)**0.5\n",
        "        self._d = torch.ones( num_params, 1, dtype=dtype, device=device) * preconditioner_init_scale\n",
        "        self._m = None # momentum buffer \n",
        "        defaults = dict(lr=lr_params)\n",
        "        super(UVd, self).__init__(self._params_with_grad, defaults)            \n",
        "\n",
        "\n",
        "    @torch.no_grad()\n",
        "    def step(self, closure):\n",
        "        \"\"\"\n",
        "        Performs a single step of PSGD with low-rank approximation (UVd) preconditioner, i.e., \n",
        "        updating the trainable parameters once, and returning what closure returns.\n",
        "\n",
        "        Args:\n",
        "            closure (callable): a closure that evaluates the function of self._params_with_grad,\n",
        "                                and returns the loss, or an iterable with the first one being loss.\n",
        "                                Random numbers, if any, used inside the closure should be generated starting \n",
        "                                from the same rng state if self.exact_hessian_vector_product = False; otherwise doesn't matter. \n",
        "        \"\"\"\n",
        "        if torch.rand([]) < self.preconditioner_update_probability:\n",
        "            # evaluates gradients, Hessian-vector product, and updates the preconditioner\n",
        "            if self.exact_hessian_vector_product:\n",
        "                # exact Hessian-vector product\n",
        "                with torch.enable_grad():\n",
        "                    closure_returns = closure()\n",
        "                    loss = closure_returns if isinstance(closure_returns, torch.Tensor) else closure_returns[0]\n",
        "                    grads = torch.autograd.grad(loss, self._params_with_grad, create_graph=True)\n",
        "                    vs = [torch.randn_like(param) for param in self._params_with_grad]\n",
        "                    Hvs = torch.autograd.grad(grads, self._params_with_grad, vs)\n",
        "            else:\n",
        "                # approximate Hessian-vector product via finite-difference formulae. Use it with cautions.\n",
        "                with torch.enable_grad():\n",
        "                    closure_returns = closure()\n",
        "                    loss = closure_returns if isinstance(closure_returns, torch.Tensor) else closure_returns[0]\n",
        "                    grads = torch.autograd.grad(loss, self._params_with_grad)\n",
        "                if self.directional == 1:\n",
        "                  vs = [self._delta_param_scale * torch.randn_like(param) for param in self._params_with_grad]\n",
        "                else:\n",
        "                  vs = [self._delta_param_scale * (1 + torch.abs(param)) * torch.randn_like(param) for param in self._params_with_grad]\n",
        "                [param.add_(v) for (param, v) in zip(self._params_with_grad, vs)]\n",
        "                with torch.enable_grad():\n",
        "                    perturbed_returns = closure()\n",
        "                    perturbed_loss = perturbed_returns if isinstance(perturbed_returns, torch.Tensor) else perturbed_returns[0]\n",
        "                    perturbed_grads = torch.autograd.grad(perturbed_loss, self._params_with_grad)\n",
        "                Hvs = [perturbed_g - g for (perturbed_g, g) in zip(perturbed_grads, grads)]\n",
        "            # update preconditioner\n",
        "            v = torch.cat([torch.flatten(v) for v in vs])\n",
        "            h = torch.cat([torch.flatten(h) for h in Hvs])\n",
        "            if self.exact_hessian_vector_product:\n",
        "                update_precond_UVd_math_(self._U, self._V, self._d,\n",
        "                                         v[:,None], h[:,None], step=self.lr_preconditioner, tiny=self._tiny)\n",
        "            else: # compensate the levels of v and h; helpful to reduce numerical errors in half-precision training\n",
        "                update_precond_UVd_math_(self._U, self._V, self._d,\n",
        "                                         v[:,None]/self._delta_param_scale, h[:,None]/self._delta_param_scale, step=self.lr_preconditioner, tiny=self._tiny)\n",
        "        else:\n",
        "            # only evaluates the gradients\n",
        "            with torch.enable_grad():\n",
        "                closure_returns = closure()\n",
        "                loss = closure_returns if isinstance(closure_returns, torch.Tensor) else closure_returns[0]\n",
        "                grads = torch.autograd.grad(loss, self._params_with_grad)\n",
        "            vs = None # no vs and Hvs\n",
        "\n",
        "        # preconditioned gradients; momentum is optional\n",
        "        grad = torch.cat([torch.flatten(g) for g in grads])\n",
        "        if self.momentum > 0:\n",
        "            if self._m is None:\n",
        "                self._m = (1 - self.momentum)*grad\n",
        "            else:\n",
        "                self._m.mul_(self.momentum).add_((1 - self.momentum)*grad)\n",
        "            pre_grad = precond_grad_UVd_math(self._U, self._V, self._d, self._m[:, None])\n",
        "        else:\n",
        "            self._m = None # clean the buffer when momentum is set to zero \n",
        "            pre_grad = precond_grad_UVd_math(self._U, self._V, self._d, grad[:, None])\n",
        "            \n",
        "        # gradient clipping is optional\n",
        "        if self.grad_clip_max_norm is None:\n",
        "            lr = self.lr_params\n",
        "        else:\n",
        "            grad_norm = torch.linalg.vector_norm(pre_grad) + self._tiny\n",
        "            lr = self.lr_params * min(self.grad_clip_max_norm/grad_norm, 1.0)\n",
        "            \n",
        "        # update the parameters\n",
        "        if self.exact_hessian_vector_product or (vs is None):\n",
        "            [param.subtract_(lr * pre_grad[j - i:j].view_as(param))\n",
        "             for (param, i, j) in zip(self._params_with_grad, self._param_sizes, self._param_cumsizes)]\n",
        "        else: # in this case, do not forget to remove the perturbation on parameters\n",
        "            [param.subtract_(lr * pre_grad[j - i:j].view_as(param) + v)\n",
        "             for (param, i, j, v) in zip(self._params_with_grad, self._param_sizes, self._param_cumsizes, vs)]\n",
        "        # return whatever closure returns\n",
        "        return closure_returns\n",
        "\n",
        "################## end of UVd preconditioner #################################\n",
        "\n",
        "\n",
        "##############################################################################\n",
        "# An Xmat (X-matrix) preconditioner is defined by\n",
        "#\n",
        "#   Q = diag(a) + adiag(b)\n",
        "#\n",
        "# where adiag means anti-diagonal.\n",
        "# It's slightly more complicated than a diagonal preconditioner, but performs better.\n",
        "#\n",
        "\n",
        "#@torch.jit.script\n",
        "def update_precond_Xmat_math_(a, b, v, h, step, tiny):\n",
        "    # type: (Tensor, Tensor, Tensor, Tensor, float, float) -> None\n",
        "    \"\"\"\n",
        "    Update preconditioner Q = diag(a) + adiag(b) with (vector, Hessian-vector product) = (v, h).\n",
        "    State variables a and b are updated inplace.\n",
        "    \"\"\"\n",
        "    Qh = a*h + b*torch.flip(h, [0])\n",
        "    aflip, bflip = torch.flip(a, [0]), torch.flip(b, [0])\n",
        "    invQtv = (aflip*v - bflip*torch.flip(v, [0]))/(a*aflip - b*bflip)\n",
        "    nablaA = Qh*Qh - invQtv*invQtv\n",
        "    nablaB = Qh*torch.flip(Qh, [0]) - invQtv*torch.flip(invQtv, [0])\n",
        "    q, r = divmod(len(nablaB), 2)\n",
        "    if r == 1:\n",
        "        nablaB[q] = 0\n",
        "\n",
        "    mu = step/(torch.maximum(torch.max(torch.abs(nablaA)), torch.max(torch.abs(nablaB))) + tiny)\n",
        "    a.sub_(mu*(nablaA*a + nablaB*bflip))\n",
        "    b.sub_(mu*(nablaA*b + nablaB*aflip))\n",
        "\n",
        "#@torch.jit.script\n",
        "def precond_grad_Xmat_math(a, b, g):\n",
        "    # type: (Tensor, Tensor, Tensor) -> Tensor\n",
        "    \"\"\"\n",
        "    Preconditioning gradient g with Q = diag(a) + adiag(b).\n",
        "    \"\"\"\n",
        "    ab = a * b\n",
        "    return (a*a + torch.flip(b*b, [0]))*g + (ab + torch.flip(ab, [0]))*torch.flip(g, [0])\n",
        "\n",
        "\n",
        "def update_precond_XMat(a,b, vs, hs, step=0.01, _tiny=1.2e-38):\n",
        "    # type: (Tensor, Tensor, List[Tensor], List[Tensor], float, float) -> Tensor\n",
        "    \"\"\"\n",
        "    update XMat preconditioner Q = diag(a) + adiag(b) with\n",
        "    vs: a list of vectors;\n",
        "    hs: a list of associated Hessian-vector products;\n",
        "    step: step size, setting to larger values, say 0.1, if updating is sparse;\n",
        "    _tiny: an offset to avoid divided by zero. \n",
        "    \n",
        "    It is a wrapped version of function update_precond_XMat_math for easy use. \n",
        "    \"\"\" \n",
        "\n",
        "    v = torch.cat([torch.flatten(v) for v in vs])\n",
        "    h = torch.cat([torch.flatten(h) for h in hs])\n",
        "    update_precond_Xmat_math_(a, b, v, h, step=step, tiny=_tiny)       \n",
        "    return a, b\n",
        "\n",
        "## Functional form of XMat Precond\n",
        "#@torch.jit.script\n",
        "def precond_grad_XMat(a,b, grads):\n",
        "\n",
        "    \"\"\"\n",
        "    return preconditioned gradient with UVd preconditioner Q = (I + U*V')*diag(d),\n",
        "    and a list of gradients, grads.\n",
        "    \n",
        "    It is a wrapped version of function precond_grad_UVd_math for easy use.\n",
        "    Also, U, V, and d are transposed (row-major order as Python convention), and \n",
        "    packaged into one tensor.\n",
        "    \"\"\"\n",
        "\n",
        "    # record the sizes and shapes, and then flatten gradients\n",
        "    # sizes = [torch.numel(g) for g in grads]\n",
        "    # shapes = [g.shape for g in grads]\n",
        "    # cumsizes = torch.cumsum(torch.tensor(sizes), 0)\n",
        "    \n",
        "    grad = torch.cat([torch.flatten(g) for g in grads])\n",
        "\n",
        "    # precondition gradients\n",
        "    pre_grad =  precond_grad_Xmat_math(a, b, grad)\n",
        "    # restore gradients to their original shapes\n",
        "    # return [torch.reshape(pre_grad[j-i:j], s) for (i, j, s) in zip(sizes, cumsizes, shapes)]\n",
        "    return torch.reshape(pre_grad,grads.shape)\n",
        "\n",
        "\n",
        "class XMat(Optimizer):\n",
        "    \"\"\"\n",
        "    Implements the Xmat preconditioner, Q = diag(a) + adiag(b), as a class.\n",
        "    Args for initialization:\n",
        "        params_with_grad: a list of parameters or variables requiring gradients;\n",
        "        preconditioner_init_scale: initial scale of Q, i.e., Q = preconditioner_init_scale*eye();\n",
        "        lr_params: normalized learning rate for parameters in range [0, 1];\n",
        "        lr_preconditioner: normalized learning rate for preconditioner in range [0, 1];\n",
        "        momentum: momentum factor in range [0,1);\n",
        "        grad_clip_max_norm: maximum allowable gradient norm after clipping, None for no clipping;\n",
        "        preconditioner_update_probability: probability on updating Q, 1 for updating at every step, and 0 for never, i.e., SGD;\n",
        "        exact_hessian_vector_product: True for exact Hessian-vector product via 2nd derivative,\n",
        "                                    and False for approximate one via finite-difference formulae.\n",
        "    Notes:\n",
        "        Note 1: The Hessian-vector product can be approximated using the finite-difference formulae by setting\n",
        "        exact_hessian_vector_product = False when the 2nd derivatives is not available.\n",
        "        In this case, make sure that the closure produces the same outputs given the same inputs,\n",
        "        except for numerical errors due to non-deterministic behaviors.\n",
        "        Random numbers, if any, used inside the closure should be generated starting from the same state, where the rng state can be\n",
        "        read and set by, e.g., `torch.cuda.get_rng_state' and `torch.cuda.set_rng_state', respectively.\n",
        "        \n",
        "        Note 2: Momentum here is the moving average of gradient so that its setting is decoupled from the learning rate.\n",
        "        This is necessary as the learning rate in PSGD is normalized.\n",
        "\n",
        "        Note 3: Currently, no support of sparse and mixed-precision gradients.\n",
        "\n",
        "        Note 4: lr_params, lr_preconditioner, momentum, grad_clip_max_norm, preconditioner_update_probability, and\n",
        "        exact_hessian_vector_product (bool) all can be reset on the fly.\n",
        "    \"\"\"\n",
        "    def __init__(self, params_with_grad, preconditioner_init_scale=1.0,\n",
        "                 lr_params=0.01, lr_preconditioner=0.01, momentum=0.0, \n",
        "                 grad_clip_max_norm=None, preconditioner_update_probability=1.0,\n",
        "                 exact_hessian_vector_product: bool = True):\n",
        "        # mutable members\n",
        "        self.lr_params = lr_params\n",
        "        self.lr_preconditioner = lr_preconditioner\n",
        "        self.momentum = momentum if (0<momentum<1) else 0.0\n",
        "        self.grad_clip_max_norm = grad_clip_max_norm\n",
        "        self.preconditioner_update_probability = preconditioner_update_probability\n",
        "        self.exact_hessian_vector_product = exact_hessian_vector_product\n",
        "        # protected members\n",
        "        params_with_grad = [params_with_grad, ] if isinstance(params_with_grad, torch.Tensor) else params_with_grad\n",
        "        self._params_with_grad = [param for param in params_with_grad if param.requires_grad]  # double check requires_grad flag\n",
        "        dtype, device = self._params_with_grad[0].dtype, self._params_with_grad[0].device\n",
        "        self._tiny = torch.finfo(dtype).tiny\n",
        "        self._delta_param_scale = torch.finfo(dtype).eps ** 0.5\n",
        "        self._param_sizes = [torch.numel(param) for param in self._params_with_grad]\n",
        "        self._param_cumsizes = torch.cumsum(torch.tensor(self._param_sizes), 0)\n",
        "        num_params = self._param_cumsizes[-1]\n",
        "        self._a = torch.ones(num_params, dtype=dtype, device=device)*preconditioner_init_scale\n",
        "        self._b = torch.zeros(num_params, dtype=dtype, device=device)\n",
        "        self._m = None # buffer for momentum \n",
        "        defaults = dict(lr=lr_params)\n",
        "        super(XMat, self).__init__(self._params_with_grad, defaults)        \n",
        "\n",
        "    @torch.no_grad()\n",
        "    def step(self, closure):\n",
        "        \"\"\"\n",
        "        Performs a single step of PSGD with Xmat preconditioner, i.e.,\n",
        "        updating the trainable parameters once, and returning what closure returns.\n",
        "        Args:\n",
        "            closure (callable): a closure that evaluates the function of self._params_with_grad,\n",
        "                                and returns the loss, or an iterable with the first one being loss.\n",
        "                                Random numbers, if any, used inside the closure should be generated starting\n",
        "                                from the same rng state if self.exact_hessian_vector_product = False; otherwise doesn't matter.\n",
        "        \"\"\"\n",
        "        if torch.rand([]) < self.preconditioner_update_probability:\n",
        "            # evaluates gradients, Hessian-vector product, and updates the preconditioner\n",
        "            if self.exact_hessian_vector_product:\n",
        "                # exact Hessian-vector product\n",
        "                with torch.enable_grad():\n",
        "                    closure_returns = closure()\n",
        "                    loss = closure_returns if isinstance(closure_returns, torch.Tensor) else closure_returns[0]\n",
        "                    grads = torch.autograd.grad(loss, self._params_with_grad, create_graph=True)\n",
        "                    vs = [torch.randn_like(param) for param in self._params_with_grad]\n",
        "                    Hvs = torch.autograd.grad(grads, self._params_with_grad, vs)\n",
        "            else:\n",
        "                # approximate Hessian-vector product via finite-difference formulae. Use it with cautions.\n",
        "                with torch.enable_grad():\n",
        "                    closure_returns = closure()\n",
        "                    loss = closure_returns if isinstance(closure_returns, torch.Tensor) else closure_returns[0]\n",
        "                    grads = torch.autograd.grad(loss, self._params_with_grad)\n",
        "                vs = [self._delta_param_scale * torch.randn_like(param) for param in self._params_with_grad]\n",
        "                [param.add_(v) for (param, v) in zip(self._params_with_grad, vs)]\n",
        "                with torch.enable_grad():\n",
        "                    perturbed_returns = closure()\n",
        "                    perturbed_loss = perturbed_returns if isinstance(perturbed_returns, torch.Tensor) else perturbed_returns[0]\n",
        "                    perturbed_grads = torch.autograd.grad(perturbed_loss, self._params_with_grad)\n",
        "                Hvs = [perturbed_g - g for (perturbed_g, g) in zip(perturbed_grads, grads)]\n",
        "            # update preconditioner\n",
        "            v = torch.cat([torch.flatten(v) for v in vs])\n",
        "            h = torch.cat([torch.flatten(h) for h in Hvs])\n",
        "            if self.exact_hessian_vector_product:\n",
        "                update_precond_Xmat_math_(self._a, self._b,\n",
        "                                         v, h, step=self.lr_preconditioner, tiny=self._tiny)\n",
        "            else:  # compensate the levels of v and h; helpful to reduce numerical errors in half-precision training\n",
        "                update_precond_Xmat_math_(self._a, self._b,\n",
        "                                         v/self._delta_param_scale, h/self._delta_param_scale,\n",
        "                                         step=self.lr_preconditioner, tiny=self._tiny)\n",
        "        else:\n",
        "            # only evaluates the gradients\n",
        "            with torch.enable_grad():\n",
        "                closure_returns = closure()\n",
        "                loss = closure_returns if isinstance(closure_returns, torch.Tensor) else closure_returns[0]\n",
        "                grads = torch.autograd.grad(loss, self._params_with_grad)\n",
        "            vs = None  # no vs and Hvs\n",
        "\n",
        "        # preconditioned gradients; momentum is optional        \n",
        "        grad = torch.cat([torch.flatten(g) for g in grads])\n",
        "        if self.momentum > 0:\n",
        "            if self._m is None:\n",
        "                self._m = (1 - self.momentum)*grad\n",
        "            else:\n",
        "                self._m.mul_(self.momentum).add_((1 - self.momentum)*grad)\n",
        "            pre_grad = precond_grad_Xmat_math(self._a, self._b, self._m)\n",
        "        else:\n",
        "            self._m = None # clean the buffer when momentum is set to zero again \n",
        "            pre_grad = precond_grad_Xmat_math(self._a, self._b, grad)\n",
        "        \n",
        "        # gradient clipping is optional\n",
        "        if self.grad_clip_max_norm is None:\n",
        "            lr = self.lr_params\n",
        "        else:\n",
        "            grad_norm = torch.linalg.vector_norm(pre_grad) + self._tiny\n",
        "            lr = self.lr_params * min(self.grad_clip_max_norm / grad_norm, 1.0)\n",
        "\n",
        "        # update the parameters\n",
        "        if self.exact_hessian_vector_product or (vs is None):\n",
        "            [param.subtract_(lr * pre_grad[j - i:j].view_as(param))\n",
        "             for (param, i, j) in zip(self._params_with_grad, self._param_sizes, self._param_cumsizes)]\n",
        "        else:  # in this case, do not forget to remove the perturbation on parameters\n",
        "            [param.subtract_(lr * pre_grad[j - i:j].view_as(param) + v)\n",
        "             for (param, i, j, v) in zip(self._params_with_grad, self._param_sizes, self._param_cumsizes, vs)]\n",
        "        # return whatever closure returns\n",
        "        return closure_returns\n",
        "\n",
        "################## end of Xmat preconditioner #################################\n",
        "\n",
        "\n",
        "###############################################################################\n",
        "# The classic Newton–Raphson type preconditioner.\n",
        "# Clearly, it is applicable only to small scale problems \n",
        "#\n",
        "\n",
        "# @torch.jit.script\n",
        "def update_precond_newton_math_(Q, v, h, step, tiny):\n",
        "    # type: (Tensor, Tensor, Tensor, float, float) -> None\n",
        "    \"\"\"\n",
        "    Update the classic Newton–Raphson type preconditioner P = Q'*Q with (v, h).\n",
        "    \"\"\"\n",
        "    a = Q.mm(h)\n",
        "    b = torch.linalg.solve_triangular(Q.t(), v, upper=False)\n",
        "    grad = torch.triu(a.mm(a.t()) - b.mm(b.t()))\n",
        "    mu = step/(grad.abs().max() + tiny)      \n",
        "    Q.sub_(mu*grad.mm(Q))\n",
        "\n",
        "class Newton:\n",
        "    \"\"\"\n",
        "    Implements the classic Newton–Raphson type preconditioner for SGD as a class.\n",
        "    Args for initialization:\n",
        "        params_with_grad: a list of parameters or variables requiring gradients;\n",
        "        preconditioner_init_scale: initial scale of Q, i.e., Q = preconditioner_init_scale*eye();\n",
        "        lr_params: normalized learning rate for parameters in range [0, 1];\n",
        "        lr_preconditioner: normalized learning rate for preconditioner in range [0, 1];\n",
        "        momentum: momentum factor in range [0,1);\n",
        "        grad_clip_max_norm: maximum allowable gradient norm after clipping, None for no clipping;\n",
        "        preconditioner_update_probability: probability on updating Q, 1 for updating at every step, and 0 for never, i.e., SGD;\n",
        "        exact_hessian_vector_product: True for exact Hessian-vector product via 2nd derivative,\n",
        "                                    and False for approximate one via finite-difference formulae.\n",
        "    Notes:\n",
        "        Note 1: The Hessian-vector product can be approximated using the finite-difference formulae by setting\n",
        "        exact_hessian_vector_product = False when the 2nd derivatives is not available.\n",
        "        In this case, make sure that the closure produces the same outputs given the same inputs,\n",
        "        except for numerical errors due to non-deterministic behaviors.\n",
        "        Random numbers, if any, used inside the closure should be generated starting from the same state, where the rng state can be\n",
        "        read and set by, e.g., `torch.cuda.get_rng_state' and `torch.cuda.set_rng_state', respectively.\n",
        "        \n",
        "        Note 2: Momentum here is the moving average of gradient so that its setting is decoupled from the learning rate.\n",
        "        This is necessary as the learning rate in PSGD is normalized.\n",
        "        Note 3: Currently, no support of sparse and mixed-precision gradients.\n",
        "        Note 4: lr_params, lr_preconditioner, momentum, grad_clip_max_norm, preconditioner_update_probability, and\n",
        "        exact_hessian_vector_product (bool) all can be reset on the fly.\n",
        "    \"\"\"\n",
        "    def __init__(self, params_with_grad, preconditioner_init_scale=1.0,\n",
        "                 lr_params=0.01, lr_preconditioner=0.01, momentum=0.0, \n",
        "                 grad_clip_max_norm=None, preconditioner_update_probability=1.0,\n",
        "                 exact_hessian_vector_product: bool = True):\n",
        "        # mutable members\n",
        "        self.lr_params = lr_params\n",
        "        self.lr_preconditioner = lr_preconditioner\n",
        "        self.momentum = momentum if (0<momentum<1) else 0.0\n",
        "        self.grad_clip_max_norm = grad_clip_max_norm\n",
        "        self.preconditioner_update_probability = preconditioner_update_probability\n",
        "        self.exact_hessian_vector_product = exact_hessian_vector_product\n",
        "        # protected members\n",
        "        params_with_grad = [params_with_grad, ] if isinstance(params_with_grad, torch.Tensor) else params_with_grad\n",
        "        self._params_with_grad = [param for param in params_with_grad if param.requires_grad]  # double check requires_grad flag\n",
        "        dtype, device = self._params_with_grad[0].dtype, self._params_with_grad[0].device\n",
        "        self._tiny = torch.finfo(dtype).tiny\n",
        "        self._delta_param_scale = torch.finfo(dtype).eps ** 0.5\n",
        "        self._param_sizes = [torch.numel(param) for param in self._params_with_grad]\n",
        "        self._param_cumsizes = torch.cumsum(torch.tensor(self._param_sizes), 0)\n",
        "        num_params = self._param_cumsizes[-1]\n",
        "        self._Q = torch.eye(num_params, dtype=dtype, device=device)*preconditioner_init_scale\n",
        "        self._m = None # buffer for momentum \n",
        "\n",
        "    @torch.no_grad()\n",
        "    def step(self, closure):\n",
        "        \"\"\"\n",
        "        Performs a single step of PSGD with Newton–Raphson preconditioner, i.e.,\n",
        "        updating the trainable parameters once, and returning what closure returns.\n",
        "        Args:\n",
        "            closure (callable): a closure that evaluates the function of self._params_with_grad,\n",
        "                                and returns the loss, or an iterable with the first one being loss.\n",
        "                                Random numbers, if any, used inside the closure should be generated starting\n",
        "                                from the same rng state if self.exact_hessian_vector_product = False; otherwise doesn't matter.\n",
        "        \"\"\"\n",
        "        if torch.rand([]) < self.preconditioner_update_probability:\n",
        "            # evaluates gradients, Hessian-vector product, and updates the preconditioner\n",
        "            if self.exact_hessian_vector_product:\n",
        "                # exact Hessian-vector product\n",
        "                with torch.enable_grad():\n",
        "                    closure_returns = closure()\n",
        "                    loss = closure_returns if isinstance(closure_returns, torch.Tensor) else closure_returns[0]\n",
        "                    grads = torch.autograd.grad(loss, self._params_with_grad, create_graph=True)\n",
        "                    vs = [torch.randn_like(param) for param in self._params_with_grad]\n",
        "                    Hvs = torch.autograd.grad(grads, self._params_with_grad, vs)\n",
        "            else:\n",
        "                # approximate Hessian-vector product via finite-difference formulae. Use it with cautions.\n",
        "                with torch.enable_grad():\n",
        "                    closure_returns = closure()\n",
        "                    loss = closure_returns if isinstance(closure_returns, torch.Tensor) else closure_returns[0]\n",
        "                    grads = torch.autograd.grad(loss, self._params_with_grad)\n",
        "                vs = [self._delta_param_scale * torch.randn_like(param) for param in self._params_with_grad]\n",
        "                [param.add_(v) for (param, v) in zip(self._params_with_grad, vs)]\n",
        "                with torch.enable_grad():\n",
        "                    perturbed_returns = closure()\n",
        "                    perturbed_loss = perturbed_returns if isinstance(perturbed_returns, torch.Tensor) else perturbed_returns[0]\n",
        "                    perturbed_grads = torch.autograd.grad(perturbed_loss, self._params_with_grad)\n",
        "                Hvs = [perturbed_g - g for (perturbed_g, g) in zip(perturbed_grads, grads)]\n",
        "            # update preconditioner\n",
        "            v = torch.cat([torch.flatten(v) for v in vs])\n",
        "            h = torch.cat([torch.flatten(h) for h in Hvs])\n",
        "            if self.exact_hessian_vector_product:\n",
        "                update_precond_newton_math_(self._Q,\n",
        "                                            v[:,None], h[:,None], step=self.lr_preconditioner, tiny=self._tiny)\n",
        "            else:  # compensate the levels of v and h; helpful to reduce numerical errors in half-precision training\n",
        "                update_precond_newton_math_(self._Q,\n",
        "                                            v[:,None]/self._delta_param_scale, h[:,None]/self._delta_param_scale,\n",
        "                                            step=self.lr_preconditioner, tiny=self._tiny)\n",
        "        else:\n",
        "            # only evaluates the gradients\n",
        "            with torch.enable_grad():\n",
        "                closure_returns = closure()\n",
        "                loss = closure_returns if isinstance(closure_returns, torch.Tensor) else closure_returns[0]\n",
        "                grads = torch.autograd.grad(loss, self._params_with_grad)\n",
        "            vs = None  # no vs and Hvs\n",
        "\n",
        "        # preconditioned gradients; momentum is optional        \n",
        "        grad = torch.cat([torch.flatten(g) for g in grads])\n",
        "        if self.momentum > 0:\n",
        "            if self._m is None:\n",
        "                self._m = (1 - self.momentum)*grad\n",
        "            else:\n",
        "                self._m.mul_(self.momentum).add_((1 - self.momentum)*grad)\n",
        "            pre_grad = self._Q.t() @ (self._Q @ self._m)\n",
        "        else:\n",
        "            self._m = None # clean the buffer when momentum is set to zero again \n",
        "            pre_grad = self._Q.t() @ (self._Q @ grad)\n",
        "        \n",
        "        # gradient clipping is optional\n",
        "        if self.grad_clip_max_norm is None:\n",
        "            lr = self.lr_params\n",
        "        else:\n",
        "            grad_norm = torch.linalg.vector_norm(pre_grad) + self._tiny\n",
        "            lr = self.lr_params * min(self.grad_clip_max_norm / grad_norm, 1.0)\n",
        "\n",
        "        # update the parameters\n",
        "        if self.exact_hessian_vector_product or (vs is None):\n",
        "            [param.subtract_(lr * pre_grad[j - i:j].view_as(param))\n",
        "             for (param, i, j) in zip(self._params_with_grad, self._param_sizes, self._param_cumsizes)]\n",
        "        else:  # in this case, do not forget to remove the perturbation on parameters\n",
        "            [param.subtract_(lr * pre_grad[j - i:j].view_as(param) + v)\n",
        "             for (param, i, j, v) in zip(self._params_with_grad, self._param_sizes, self._param_cumsizes, vs)]\n",
        "        # return whatever closure returns\n",
        "        return closure_returns\n",
        "\n",
        "################## end of Newton–Raphson preconditioner #################################"
      ],
      "metadata": {
        "id": "-Ssga-nW9fNk",
        "cellView": "form"
      },
      "execution_count": 1,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Models"
      ],
      "metadata": {
        "id": "8uj_9DcLrZfO"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Generate Data"
      ],
      "metadata": {
        "id": "qeq7aDhdr0fD"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "#@title RNN network with the classic delayed XOR problem. Accelerated directional derivative \n",
        "\n",
        "import matplotlib.pyplot as plt\n",
        "import numpy as np\n",
        "import torch\n",
        "# import preconditioned_stochastic_gradient_descent as psgd\n",
        "\n",
        "device = torch.device('cpu')\n",
        "batch_size, seq_len = 128, 32          # increasing sequence_length or decreasing dimension_hidden_layer will make learning harder;\n",
        "dim_in, dim_hidden, dim_out = 2, 30, 1  # current setting can solve seq len 80 ~ 90 reliably without the help of momentum \n",
        "\n",
        "def generate_train_data():\n",
        "    x = np.zeros([batch_size, seq_len, dim_in], dtype=np.float32)\n",
        "    y = np.zeros([batch_size, dim_out], dtype=np.float32)\n",
        "    for i in range(batch_size):\n",
        "        x[i, :, 0] = np.random.choice([-1.0, 1.0], seq_len)\n",
        "\n",
        "        i1 = int(np.floor(np.random.rand() * 0.1 * seq_len))\n",
        "        i2 = int(np.floor(np.random.rand() * 0.4 * seq_len + 0.1 * seq_len))\n",
        "        x[i, i1, 1] = 1.0\n",
        "        x[i, i2, 1] = 1.0\n",
        "        if x[i, i1, 0] == x[i, i2, 0]:  # XOR\n",
        "            y[i] = -1.0  # lable 0\n",
        "        else:\n",
        "            y[i] = 1.0  # lable 1\n",
        "\n",
        "    # tranpose x to format (sequence_length, batch_size, dimension_of_input)\n",
        "    return [torch.tensor(np.transpose(x, [1, 0, 2])).to(device),\n",
        "            torch.tensor(y).to(device)]\n",
        "\n",
        "# generate a random orthogonal matrix for recurrent matrix initialization\n",
        "def get_rand_orth(dim):\n",
        "    temp = np.random.normal(size=[dim, dim])\n",
        "    q, _ = np.linalg.qr(temp)\n",
        "    return torch.tensor(q, dtype=torch.float32).to(device)\n",
        "\n",
        "class Model(torch.nn.Module):\n",
        "    def __init__(self):\n",
        "        super(Model, self).__init__()\n",
        "        self.W1x = torch.nn.Parameter(0.1 * torch.randn(dim_in, dim_hidden))\n",
        "        self.W1h = torch.nn.Parameter(get_rand_orth(dim_hidden))\n",
        "        self.b1 = torch.nn.Parameter(torch.zeros(dim_hidden))\n",
        "        self.W2 = torch.nn.Parameter(0.1 * torch.randn(dim_hidden, dim_out))\n",
        "        self.b2 = torch.nn.Parameter(torch.zeros([]))\n",
        "\n",
        "    def forward(self, xs):\n",
        "        h = torch.zeros(batch_size, dim_hidden, device=device)\n",
        "        for x in torch.unbind(xs):\n",
        "            h = torch.tanh(x @ self.W1x + h @ self.W1h + self.b1)\n",
        "        return h @ self.W2 + self.b2\n",
        "\n",
        "model = Model().to(device)\n",
        "# initialize the PSGD optimizer with the low-rank approximation preconditioner \n",
        "opt = UVd(model.parameters(),\n",
        "               # rank_of_approximation=10, preconditioner_init_scale=1.0,\n",
        "               # lr_params=0.01, lr_preconditioner=0.01, momentum=0.9,\n",
        "               grad_clip_max_norm=1.0, # preconditioner_update_probability=1.0,\n",
        "               exact_hessian_vector_product=False,\n",
        "               momentum=0.9,\n",
        "               directional = 0\n",
        "               )\n",
        "\n",
        "\n",
        "def train_loss(xy_pair):  # logistic loss\n",
        "    return -torch.mean(torch.log(torch.sigmoid(xy_pair[1] * model(xy_pair[0]))))\n",
        "\n",
        "Losses = []\n",
        "for num_iter in range(10000):\n",
        "    train_data = generate_train_data()\n",
        "\n",
        "    rng_state = torch.get_rng_state()\n",
        "    # rng_cuda_state = torch.cuda.get_rng_state()\n",
        "    def closure(): \n",
        "        # If exact_hessian_vector_product=False and rng is used inside closure, \n",
        "        # make sure rng starts from the same state; otherwise, doesn't matter. \n",
        "        torch.set_rng_state(rng_state)\n",
        "        # torch.cuda.set_rng_state(rng_cuda_state)\n",
        "        return train_loss(train_data) # return a loss\n",
        "\n",
        "    loss = opt.step(closure)\n",
        "    Losses.append(loss.item())\n",
        "    print('Iteration: {}; loss: {}'.format(num_iter, Losses[-1]))\n",
        "\n",
        "plt.plot(Losses)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 1000
        },
        "id": "SoxKl2JUHxsi",
        "outputId": "b0441b73-a362-4621-e93b-2990136f4b7a"
      },
      "execution_count": 5,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "\u001b[1;30;43mStreaming output truncated to the last 5000 lines.\u001b[0m\n",
            "Iteration: 5000; loss: 0.6909657120704651\n",
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            "Iteration: 9948; loss: 0.07384231686592102\n",
            "Iteration: 9949; loss: 0.022164393216371536\n",
            "Iteration: 9950; loss: 0.03868719935417175\n",
            "Iteration: 9951; loss: 0.06663373112678528\n",
            "Iteration: 9952; loss: 0.02513979747891426\n",
            "Iteration: 9953; loss: 0.02355808950960636\n",
            "Iteration: 9954; loss: 0.02167792059481144\n",
            "Iteration: 9955; loss: 0.09090021252632141\n",
            "Iteration: 9956; loss: 0.06634268164634705\n",
            "Iteration: 9957; loss: 0.0386153906583786\n",
            "Iteration: 9958; loss: 0.03769129514694214\n",
            "Iteration: 9959; loss: 0.04862430319190025\n",
            "Iteration: 9960; loss: 0.06138962507247925\n",
            "Iteration: 9961; loss: 0.02510054036974907\n",
            "Iteration: 9962; loss: 0.018100567162036896\n",
            "Iteration: 9963; loss: 0.07045136392116547\n",
            "Iteration: 9964; loss: 0.019331667572259903\n",
            "Iteration: 9965; loss: 0.023115642368793488\n",
            "Iteration: 9966; loss: 0.01381669845432043\n",
            "Iteration: 9967; loss: 0.05048048123717308\n",
            "Iteration: 9968; loss: 0.031140757724642754\n",
            "Iteration: 9969; loss: 0.020053712651133537\n",
            "Iteration: 9970; loss: 0.014243911020457745\n",
            "Iteration: 9971; loss: 0.018186435103416443\n",
            "Iteration: 9972; loss: 0.03449749946594238\n",
            "Iteration: 9973; loss: 0.07400649785995483\n",
            "Iteration: 9974; loss: 0.012453676201403141\n",
            "Iteration: 9975; loss: 0.032029859721660614\n",
            "Iteration: 9976; loss: 0.04788685217499733\n",
            "Iteration: 9977; loss: 0.02754206210374832\n",
            "Iteration: 9978; loss: 0.01872710883617401\n",
            "Iteration: 9979; loss: 0.019876154139637947\n",
            "Iteration: 9980; loss: 0.03846703842282295\n",
            "Iteration: 9981; loss: 0.015514438971877098\n",
            "Iteration: 9982; loss: 0.045087285339832306\n",
            "Iteration: 9983; loss: 0.03148084878921509\n",
            "Iteration: 9984; loss: 0.027225568890571594\n",
            "Iteration: 9985; loss: 0.015649929642677307\n",
            "Iteration: 9986; loss: 0.020747123286128044\n",
            "Iteration: 9987; loss: 0.030103864148259163\n",
            "Iteration: 9988; loss: 0.0281609445810318\n",
            "Iteration: 9989; loss: 0.030107207596302032\n",
            "Iteration: 9990; loss: 0.022747481241822243\n",
            "Iteration: 9991; loss: 0.018807509914040565\n",
            "Iteration: 9992; loss: 0.01809319108724594\n",
            "Iteration: 9993; loss: 0.02590429224073887\n",
            "Iteration: 9994; loss: 0.013305862434208393\n",
            "Iteration: 9995; loss: 0.0418723002076149\n",
            "Iteration: 9996; loss: 0.013979995623230934\n",
            "Iteration: 9997; loss: 0.015125597827136517\n",
            "Iteration: 9998; loss: 0.017207343131303787\n",
            "Iteration: 9999; loss: 0.008721093647181988\n"
          ]
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f9c0007ef20>]"
            ]
          },
          "metadata": {},
          "execution_count": 5
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "#@title RNN network with the classic delayed XOR problem. Standard directional derivative \n",
        "\n",
        "import matplotlib.pyplot as plt\n",
        "import numpy as np\n",
        "import torch\n",
        "# import preconditioned_stochastic_gradient_descent as psgd\n",
        "\n",
        "device = torch.device('cpu')\n",
        "batch_size, seq_len = 128, 32          # increasing sequence_length or decreasing dimension_hidden_layer will make learning harder;\n",
        "dim_in, dim_hidden, dim_out = 2, 30, 1  # current setting can solve seq len 80 ~ 90 reliably without the help of momentum \n",
        "\n",
        "def generate_train_data():\n",
        "    x = np.zeros([batch_size, seq_len, dim_in], dtype=np.float32)\n",
        "    y = np.zeros([batch_size, dim_out], dtype=np.float32)\n",
        "    for i in range(batch_size):\n",
        "        x[i, :, 0] = np.random.choice([-1.0, 1.0], seq_len)\n",
        "\n",
        "        i1 = int(np.floor(np.random.rand() * 0.1 * seq_len))\n",
        "        i2 = int(np.floor(np.random.rand() * 0.4 * seq_len + 0.1 * seq_len))\n",
        "        x[i, i1, 1] = 1.0\n",
        "        x[i, i2, 1] = 1.0\n",
        "        if x[i, i1, 0] == x[i, i2, 0]:  # XOR\n",
        "            y[i] = -1.0  # lable 0\n",
        "        else:\n",
        "            y[i] = 1.0  # lable 1\n",
        "\n",
        "    # tranpose x to format (sequence_length, batch_size, dimension_of_input)\n",
        "    return [torch.tensor(np.transpose(x, [1, 0, 2])).to(device),\n",
        "            torch.tensor(y).to(device)]\n",
        "\n",
        "# generate a random orthogonal matrix for recurrent matrix initialization\n",
        "def get_rand_orth(dim):\n",
        "    temp = np.random.normal(size=[dim, dim])\n",
        "    q, _ = np.linalg.qr(temp)\n",
        "    return torch.tensor(q, dtype=torch.float32).to(device)\n",
        "\n",
        "class Model(torch.nn.Module):\n",
        "    def __init__(self):\n",
        "        super(Model, self).__init__()\n",
        "        self.W1x = torch.nn.Parameter(0.1 * torch.randn(dim_in, dim_hidden))\n",
        "        self.W1h = torch.nn.Parameter(get_rand_orth(dim_hidden))\n",
        "        self.b1 = torch.nn.Parameter(torch.zeros(dim_hidden))\n",
        "        self.W2 = torch.nn.Parameter(0.1 * torch.randn(dim_hidden, dim_out))\n",
        "        self.b2 = torch.nn.Parameter(torch.zeros([]))\n",
        "\n",
        "    def forward(self, xs):\n",
        "        h = torch.zeros(batch_size, dim_hidden, device=device)\n",
        "        for x in torch.unbind(xs):\n",
        "            h = torch.tanh(x @ self.W1x + h @ self.W1h + self.b1)\n",
        "        return h @ self.W2 + self.b2\n",
        "\n",
        "model = Model().to(device)\n",
        "# initialize the PSGD optimizer with the low-rank approximation preconditioner \n",
        "opt = UVd(model.parameters(),\n",
        "               # rank_of_approximation=10, preconditioner_init_scale=1.0,\n",
        "               # lr_params=0.01, lr_preconditioner=0.01, momentum=0.9,\n",
        "               grad_clip_max_norm=1.0, # preconditioner_update_probability=1.0,\n",
        "               exact_hessian_vector_product=False,\n",
        "               momentum=0.9,\n",
        "               directional = 1\n",
        "               )\n",
        "\n",
        "\n",
        "def train_loss(xy_pair):  # logistic loss\n",
        "    return -torch.mean(torch.log(torch.sigmoid(xy_pair[1] * model(xy_pair[0]))))\n",
        "\n",
        "Losses = []\n",
        "for num_iter in range(10000):\n",
        "    train_data = generate_train_data()\n",
        "\n",
        "    rng_state = torch.get_rng_state()\n",
        "    # rng_cuda_state = torch.cuda.get_rng_state()\n",
        "    def closure(): \n",
        "        # If exact_hessian_vector_product=False and rng is used inside closure, \n",
        "        # make sure rng starts from the same state; otherwise, doesn't matter. \n",
        "        torch.set_rng_state(rng_state)\n",
        "        # torch.cuda.set_rng_state(rng_cuda_state)\n",
        "        return train_loss(train_data) # return a loss\n",
        "\n",
        "    loss = opt.step(closure)\n",
        "    Losses.append(loss.item())\n",
        "    print('Iteration: {}; loss: {}'.format(num_iter, Losses[-1]))\n",
        "\n",
        "plt.plot(Losses)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 1000
        },
        "id": "_kyniHT-mAVt",
        "outputId": "257c0b50-d237-41fc-9823-76601d75a0fc"
      },
      "execution_count": 4,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "\u001b[1;30;43mStreaming output truncated to the last 5000 lines.\u001b[0m\n",
            "Iteration: 5000; loss: 0.6902387738227844\n",
            "Iteration: 5001; loss: 0.6918805837631226\n",
            "Iteration: 5002; loss: 0.6921588182449341\n",
            "Iteration: 5003; loss: 0.6937887072563171\n",
            "Iteration: 5004; loss: 0.6920034289360046\n",
            "Iteration: 5005; loss: 0.6938259601593018\n",
            "Iteration: 5006; loss: 0.6929646730422974\n",
            "Iteration: 5007; loss: 0.6949727535247803\n",
            "Iteration: 5008; loss: 0.6930686235427856\n",
            "Iteration: 5009; loss: 0.6940971612930298\n",
            "Iteration: 5010; loss: 0.6929494142532349\n",
            "Iteration: 5011; loss: 0.6967167854309082\n",
            "Iteration: 5012; loss: 0.6926057934761047\n",
            "Iteration: 5013; loss: 0.6938834190368652\n",
            "Iteration: 5014; loss: 0.6947521567344666\n",
            "Iteration: 5015; loss: 0.6932332515716553\n",
            "Iteration: 5016; loss: 0.6920724511146545\n",
            "Iteration: 5017; loss: 0.6916383504867554\n",
            "Iteration: 5018; loss: 0.6922935843467712\n",
            "Iteration: 5019; loss: 0.6946487426757812\n",
            "Iteration: 5020; loss: 0.6924275755882263\n",
            "Iteration: 5021; loss: 0.6899878978729248\n",
            "Iteration: 5022; loss: 0.695713996887207\n",
            "Iteration: 5023; loss: 0.6928800940513611\n",
            "Iteration: 5024; loss: 0.6913423538208008\n",
            "Iteration: 5025; loss: 0.6936554908752441\n",
            "Iteration: 5026; loss: 0.6906473636627197\n",
            "Iteration: 5027; loss: 0.6908701062202454\n",
            "Iteration: 5028; loss: 0.6922637820243835\n",
            "Iteration: 5029; loss: 0.6943365931510925\n",
            "Iteration: 5030; loss: 0.692548394203186\n",
            "Iteration: 5031; loss: 0.6931653022766113\n",
            "Iteration: 5032; loss: 0.6921069025993347\n",
            "Iteration: 5033; loss: 0.691778302192688\n",
            "Iteration: 5034; loss: 0.6903672218322754\n",
            "Iteration: 5035; loss: 0.6951153874397278\n",
            "Iteration: 5036; loss: 0.6912813186645508\n",
            "Iteration: 5037; loss: 0.6937152147293091\n",
            "Iteration: 5038; loss: 0.695313036441803\n",
            "Iteration: 5039; loss: 0.6914244294166565\n",
            "Iteration: 5040; loss: 0.6952090859413147\n",
            "Iteration: 5041; loss: 0.6951967477798462\n",
            "Iteration: 5042; loss: 0.6916675567626953\n",
            "Iteration: 5043; loss: 0.6948449611663818\n",
            "Iteration: 5044; loss: 0.6899482607841492\n",
            "Iteration: 5045; loss: 0.6956689953804016\n",
            "Iteration: 5046; loss: 0.6911846995353699\n",
            "Iteration: 5047; loss: 0.6954415440559387\n",
            "Iteration: 5048; loss: 0.6921917796134949\n",
            "Iteration: 5049; loss: 0.6914990544319153\n",
            "Iteration: 5050; loss: 0.6937078237533569\n",
            "Iteration: 5051; loss: 0.6941996812820435\n",
            "Iteration: 5052; loss: 0.6933162212371826\n",
            "Iteration: 5053; loss: 0.6938217878341675\n",
            "Iteration: 5054; loss: 0.6934221982955933\n",
            "Iteration: 5055; loss: 0.6947835087776184\n",
            "Iteration: 5056; loss: 0.69107985496521\n",
            "Iteration: 5057; loss: 0.692050039768219\n",
            "Iteration: 5058; loss: 0.6939049959182739\n",
            "Iteration: 5059; loss: 0.6952390670776367\n",
            "Iteration: 5060; loss: 0.69347083568573\n",
            "Iteration: 5061; loss: 0.6954789161682129\n",
            "Iteration: 5062; loss: 0.6940256953239441\n",
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          ]
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f9c03454be0>]"
            ]
          },
          "metadata": {},
          "execution_count": 4
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [],
      "metadata": {
        "id": "57Gx-GFTyG8g"
      },
      "execution_count": null,
      "outputs": []
    }
  ]
}